THE TOTAL NUMBER OF SUBGROUPS OF A FINITE ABELIAN GROUP. Grigore Călugăreanu. Received November 11, 2003
|
|
- Everett Collins
- 5 years ago
- Views:
Transcription
1 Scientiae Mathematicae Japonicae Online, Vol. 1, (23), TE TOTAL NMBER OF SBROPS OF A FINITE ABELIAN ROP rigore Călugăreanu Received November 11, 23 Abstract. In this note, steps in order to write a formula that gives the total number of subgroups of a finite abelian group are made. 1 Introduction It is well-known (Frobenius-Stickelberger, 1878, see [4]) that a finite abelian group is the direct sum of a finite number of cyclic groups of prime power orders. Obviously, their subgroups have the same structure. owever, when it comes to draw the subgroup lattice of a given finite abelian group, or to find out the total number of subgroups this group has, this can be a difficult task. It is the purpose of this note to describe a new method which partially solves these two problems. First of all, let us mention some reductions which bring us closer to the real problem. Let be a finite abelian group and = n = p r1 1 pr2 2...pr k k, the decomposition of its order into prime power factors. If = p1 p2... pk is the corresponding primary decomposition, then, denoting by L() the subgroup lattice of, L() L( p1 ) L( p2 )... L( pk ), the direct product of the corresponding subgroup lattices (Suzuki [11]). In the sequel we denote by N() the number of subgroups of the group. ence k N() = N( ps ) and our counting problem is reduced to p-groups. Moreover, we shall i=1 consider that, given the subgroup lattices of the primary components, one can construct, the direct product of these lattices, and this finally gives the subgroup lattice of. Therefore both problems reduce to primary groups. In the above decomposition, the subgroup lattice L() has generally more subgroups than the direct product of the subgroup lattices of the direct components. As for our first problem, we will emphasize, which are the subgroups one has to add to the direct product of the subgroup lattices of the direct components. This will be possible using the key result described in the next section. As for our second problem, we will find a formula which gives the total number of subgroups of an abelian finite group whose p-ranks do not exceed two. Formulas which give the number of subgroups of type µ of a given finite p-group of type λ were given by Delsarte (see [2]), Djubjuk (see [3]), and Yeh (see [12]) in The reader may consult an excellent survey on this topic together with connections to symmetric functions written by L. Butler (see [1]). We mention below two of these formulas. The application of the theory of symmetric functions to the study of abelian groups begins with P. all s unpublished work in the 195 s (according to Macdonald [7], these 2 Mathematics Subject Classification. 21, 227, 23. ey words and phrases. diagonal, segment, interval, subgroup lattice. This work was supported by uwait niversity Research Administration rant No. SM 3/3.
2 28 RIORE CĂLĂREAN were conjectured in 19 by Steinitz, see [1]). all proves that the number of subgroups of type µ of a finite abelian p-group of type λ such that the type of / is ν, isa polynomial in p with integer coefficients, the all polynomial g λ µν. Moreover, he finds its degree and leading coefficient and shows g λ µν = gλ νµ. Lemma 1.4.1([1]) For any partitions µ λ, the number of subgroups of type µ in a finite abelian p-group of type λ is α λ (µ; p) = [ p µ j+1 (λ j µ j+1 λ ) j µ ] j+1 µ j 1 j µ j+1 where λ is the conjugate of λ, and µ is the conjugate of µ. Remarks. -Ifλ =(λ 1,λ 2...) is a partition, then the conjugate partition λ =(λ 1,λ 2...) has components defined as follows: λ j is the number of parts λ i such that λ i j. (So the jth row of the Ferrers diagram of λ is the jth column of the Ferrers diagram of λ.) [ ] n - ere k p = k i=1 n (p i 1) i=1 n k (p i 1) (p i 1) i=1 the k-dimensional subspaces of an n-space. - The formula given by Djubjuk (see [3]) is similar. ere is a second formula (adapted from [2]): denote the aussian coefficients, the number of all p s1r2+s2r3+...s r 1 1 k 1r k (p s1 p i1 ) r2 1 (p s2 p i2 )... rk 1 (p s k p i k ) i N(X; Y )= 1=r 2 i 2=r 3 i 1= p r1r2+r2r3+...r r 1 1 k 1r k (p r1 p i1 ) r2 1 (p r2 p i2 )... rk 1 (p r k p i k) i 1=r 2 i 2=r 3 i 1= where s 1 s 2... s k 1 is the signature of a group of type Y and r 1 r 2... r k 1 is the signature of a subgroup of type X (r j s j,1 j k). With our above notation the connection between the invariants which give the type Y =(n 1,n 2,..., n k ) and the signature (s 1,s 2,..., s k ) is given by the following relations: or s 1 s 2 = n 1,s 2 s 3 = n 2,..., s k 1 s k = n k 1,s k = n k s 1 = n 1 + n n k,s 2 = n 2 + n n k,..., s k 1 = n k 1 + n k,s k = n k. p owever, for a given p-group, to sum up the numbers given by the above formulas is another difficult task (e.g., notice that for t<kthe signature of = Z p t Z p k is (2, 2,..., 2, 1, 1,..., 1) and one has to sum up all the subgroups considering all the possible sub-signatures!). That s why, our method (which seems to be original, and natural since it describes directly the corresponding subgroup lattices) gives a formula for the total number of subgroups of a (finite) abelian rank two p-group which, to the best of our knowledge, did not yet appear in print.
3 TE TOTAL NMBER OF SBROPS OF A FINITE ABELIAN ROP 29 2 The key result Our key result is antique. It was discovered by oursat as early as 189 (see [6]): it is possible to construct the subgroup lattice of a direct sum out of the subgroup lattices of the summands, and all the isomorphisms between sections! A brief presentation follows (for details see [9]). (a2) Let be a subgroup of =. Then there is natural isomorphism ( + ) ( + ). Conversely, let = be a group and W 1, W 2 subgroups of the direct summands. For every isomorphism δ : there exists a subgroup W 1 W 2 such that =( + ), =( + ), W 1 = and W 2 =, namely = D(,δ)={x + y x,y δ(x + W 1 )}. Thus in order to recover the subgroups of a direct sum we need the isomorphisms between the sections (i.e., intervals in the subgroup lattice) in [,] respectively [,]. Let =. A subgroup D of is called a diagonal in (with respect to and ) ifd + = = D + and D ==D. If δ : is an isomorphism then D(δ) =D(, δ) ={x + δ(x) x } =(1+δ)() is a diagonal in (with respect to and ). Conversely, if D is a diagonal in (with respect to and ) there is a unique isomorphism δ : such that D = D(δ). More, there is a bijection between the diagonals (with respect to and ) and isomorphisms of and. Every subgroup of a direct sum = belongs to the direct product L = L() L() (i.e., has the form for and ) orisa diagonal. The situation is best described by the following diagram
4 21 RIORE CĂLĂREAN sing the projections from the direct sum, = p (), = p (), =( ) ( ) L and =( + ) ( + ) L. Moreover (see [5] where these are described as subdirect sums), is minimal in L including and = ( ) ( ) ismaximal in L included in. Finally, this allows us to reconstruct, beginning with the direct product of the subgroup lattices L(), L() all the other subgroups, namely the diagonals, just looking at the isomorphisms between the sections. 3 Examples 1) The lein group = = Z 2 Z 2 = {,a,b,a+b 2a ==2b}. Each L(Z 2 ) is a chain with two elements and the direct product of these two chains is the four element lattice sing the key result, we have to add as many diagonals as isomorphisms Z 2 Z 2 (i.e., ). From = {,a} to = {,b} there is only one isomorphism so only one diagonal (namely (1 + δ)() ={,a+ b}) has to be added to the direct product. ence has the diamond as subgroup lattice D 2) The elementary group = Z 3 Z 3 = {,a,2a, b, 2b, a + b, a +2b, 2a + b, 2a +2b 3a = =3b}. The same two chains of two elements and four element direct product ( of lattices. ) a 2a owever, now there are two isomorphisms Z 3 Z 3, namely δ : and b 2b ( ) a 2a δ :. ence we must add two diagonals D =(1+δ)() ={,a+ b, 2a +2b} 2b b respectively D =(1+δ )() ={,a+2b, 2a + b}. The following diagram describes this D D
5 TE TOTAL NMBER OF SBROPS OF A FINITE ABELIAN ROP 211 3) The group = Z 2 Z 4 = {,a,b,a+ b, 2b, a +2b, 3b, a +3b 2a ==4b} has the subgroup lattice +(2) D D 2 Indeed, to the direct product of chains we have to add only two diagonals, D corresponding to the isomorphism [,] [, 2] respectively D to the isomorphism [,] [2, ]. 4) The group = Z 4 Z 4 = a, b 4a =4b = with cyclic subgroups N = a = 3a, M = b = 3b has the subgroup lattice N+V S+ T+M N V S T M 2N 2S 2M Once again, we start by two 3-element chains (i.e., of length 2), and the direct product has 9 elements. For the first time we have here two kind of isomorphisms: four isomorphisms Z 2 Z 2 ([, 2N] [, 2M], [, 2N] [2M, M], [2N, N] [, 2M], [2N, N] [2M, M])
6 212 RIORE CĂLĂREAN respectively two isomorphisms Z 4 Z 4 ([,N] [,M]-Z 4 has two automorphisms), so we add 6 diagonals, 2S, T, V, S + and S,, for a total of 15 subgroups. 5) The elementary group = Z 2 Z 2 Z 2 has the following subgroup lattice A B In order to recover this by using our key result, we consider = Z 2 with a 2-element chain subgroup lattice, and = Z 2 Z 2, with the diamond subgroup lattice (see Example 1). We have 2 5 = 1 subgroups in the direct product + 6 diagonals to add, corresponding to the 6 isomorphisms associated to all the segments in the diamond. That is, 16 subgroups indeed. 6) The 2-group Z 4 Z 2 Z 2. Now we take = Z 4 and = Z 2 Z 2, so a 3-element chain and the diamond as subgroup lattices, as follows B 2 a b c A ence we have 3 5 = 15 subgroups in the direct product, and we must add diagonals corresponding to 6 isomorphisms from A =[, 2] to each of the segments on the diamond and other 6 isomorphisms from B =[2, ] to the same 6 segments. That is = 27 subgroups. Remark. Apparently there would be some isomorphisms from the interval (chain) [,] to some 3-element chains in the diamond (namely [,a,], [,b,] or[,c,]). Wrong! None of these actually is an interval in the subgroup lattice L().
7 TE TOTAL NMBER OF SBROPS OF A FINITE ABELIAN ROP 213 7) = Z 2 (Z 4 Z 4 ). First, the direct product of a 2-element chain and the lattice in Example 5. Thus 2 15 = 3 subgroups in the direct product + as many diagonals as isomorphisms Z 2 Z 2. Now we have as many such isomorphisms as many segments has the lattice L(Z 4 Z 4 ); this number is 24. Finally we have = 54 subgroups. 8) = Z 8 (Z 2 Z 4 ) Apparently similar with the previous, this Example emphasizes a new aspect. First we have 4 8 = 32 subgroups in the direct product diagonals for the corresponding (simple) segments [hereafter also called 1-segments]. 2 b D 4 D a Next we must count 2-segments (on the right lattice there are no 3-segments which are intervals): 2 as (long) sides of the right rectangle and the chains (which actually are also intervals!) [,a,d ] respectively [D, b, ]. ence, = 16 more diagonals to be added (2 for [, 2] and [4, ], Aut(Z 4 ) = 2(2 1) = 2; see also next section, respectively the 4 2-segments), for a total of = 81 subgroups. 9) =(Z 2 Z 2 ) (Z 4 Z 4 ) is the only group of rank four we discuss in this Section. Consider = Z 2 Z 2 and = Z 4 Z 4 which have as subgroup lattices the diamond (Example 1) and the lattice L(Z 4 Z 4 ) from Example 5 which has 15 subgroups and 24 segments. ence we have 5 15 = 75 subgroups in the direct product, 6 24 = 144 diagonals corresponding to all the Z 2 Z 2 isomorphisms (between the 1-segments) and (for the first time), 5 isomorphisms of diamonds (indeed, L(Z 4 Z 4 ) has 5 intervals which are diamonds ). That is, 5 Aut(Z 2 Z 2 ) =5 6 = 3 (the lein group Z 2 Z 2 has 3! = 6 automorphisms). So the total is = 249 subgroups.
8 214 RIORE CĂLĂREAN 4 The rank 2 formula Diagonals, segments and automorphisms. In what follows, segment on the diagram that represents a given subgroup lattice has the usual geometric meaning and, as it is well-known, is drawn whenever a subgroup S is covered by another subgroup T (i.e., T/S is simple). Moreover, the term interval (we have finally preferred it to the synonymous section ) is used as follows: if S, T are subgroups of, the interval [S, T ]={ L() S T }. As previously noted, this is generally not a chain (excepting cocyclic groups) nor a diamond. Notice that counting segments may be done by counting the segments in the direct product and adding 2 times the number of diagonals (obviously, each diagonal comes with exactly two 1-segments - in producing the corresponding diamond ). For later purposes, the two 1-segments adjacent to a diagonal will be called half-diagonals (thus, diagonal means a subgroup, and half-diagonal means an 1-segment). Recall (e.g. see [5]) that Aut(Z n ) = ϕ(n) the Euler (arithmetic) function. Thus Aut(Z p t) = ϕ(p t )=p t (1 1 p )=pt 1 (p 1) gives the number of isomorphisms between two intervals of the form [, Z p t] (i.e., chains of length t). The formula. In what follows we shall prove the formula giving the total number of subgroups for a (finite) rank two p-group. Set = Z p t Z p k for arbitrary positive integers t and k. First of all, we have the direct product of chains of length t respectively k, that is, (t + 1)(k + 1) subgroups. Next, we have the diagonals corresponding to the automorphisms Z p Z p, which give p 1 diagonals for each pair of 1-segments, i.e., tk(p 1). Further, the diagonals corresponding to the automorphisms Z p 2 Z p 2, which give p(p 1) diagonals for each pair of (double) adjacent segments (2-segments). So (t 1)(k 1)p(p 1) diagonals have to be added. We must continue till we exhaust the adjacent min(t, k)-length segments, which obviously is the chain L() or the chain L() (corresponding to t k respectively k t). This chain produces k t + 1 pairs of chains of length min(t, k), each giving p min(t,k) 1 (p 1) diagonals. Therefore the total number of subgroups is (t + 1)(k +1)+tk(p 1) + (t 1)(k 1)p(p 1) ( k t +2)p min(t;k) 2 (p 1) + ( k t +1)p min(t;k) 1 (p 1). In what follows a special sum will be used. For t k denote by t 1 S tk = (t i)(k i)p i i=
9 TE TOTAL NMBER OF SBROPS OF A FINITE ABELIAN ROP 215 Considering the polynomial f =1+X + X X t 1 Z[X], this sum is S tk = tkf(p) (t + k)pf (p)+p[(xf (X)) ](p) (here f (X) = (t 1)X t tx t 1 +1 (X 1) 2 and (Xf (X)) = (t 1)2 X t+1 (2t 2 2t 1)X t + t 2 X t 1 X 1 (X 1) 3 ). ence S tk is calculated as follows: tk pt 1 p 1 (t+k)p(t 1)pt tp t 1 +1 (p 1) 2 +p (t 1)2 p t+1 (2t 2 2t 1)p t + t 2 p t 1 p 1 (p 1) 3, or p t+2 (k t +1)+p t+1 (t k +1) (t + 1)(k +1)p 2 +(2tk + t + k 1)p tk (p 1) 3. Moreover, for two arbitrary positive integers t, k denote by S tk = S min(t;k),max(t;k). Then we obtain N(Z p t Z p k)=(t + 1)(k +1)+(p 1)S tk or finally (t + 1)(k +1)+ + pt+2 ( k t +1)+p t+1 (1 k t ) (t + 1)(k +1)p 2 +(2tk + t + k 1)p tk (p 1) 2 =(t + 1)(k +1)+ pt+2 ( k t +1)+p t+1 (1 k t ) p [(t +1)p t][(k +1)p k] (p 1) 2. Remarks.- As for (finite) rank three p-groups, the use of the same key result is possible but complicated (the previous presented Examples suggest which are the difficulties). It will be done elsewhere. owever, subgroups of a direct sum of three groups were already described in 1931 by Remak (see [8]). Actually this is a difficult paper to read (some 156 equations describe finally the corresponding key result!). The use of the corresponding isomorphism (for direct sum of two groups this was (a2) in our second Section) is more effective in giving the formula for the total number of subgroups for a rank three p-group: (a3) Let = and be a subgroup of. There is natural isomorphism using the notations: (L 1 L 2 L 3 )+ L 1 L 2 L 3 = N 1 + N 2 + N 3 B 1 = 1 ( +( 2 3 )), A 12 =( + 1 ) 2, 1 =( + 1 ) ( 2 3 ), C 1 = 1, D 1 = ( 2 3 ), L 1 = A 12 A 13, M 1 = A 12 + A 13, N 1 = D 1 (A 12 A 13 ), and the other combinations interchanging indexes 1,2,3.
10 216 RIORE CĂLĂREAN One has to write the converse (as for the two summands case) and to establish the corresponding bijection between subgroups and isomorphism of intervals. This gives, in a similar way, the total number of subgroups. 5 Elementary abelian groups; recurrences. Any finite elementary p-group, is represented as Z n p = Z p n for a positive integer n. Denote by N(Z n p ) its total number of subgroups and by n(1 seg(l(z n p ))) the number of 1-segments the subgroup lattice L(Z n p ) has. sing the aussian coefficients, one can n [ ] n define the alois numbers n,p =. Recall that for these numbers there is only k k= p a recursion (and no formula): for all N N and p prime (power),p =1, 1,p =2, n+1,p =2 n,p +(p n 1) n 1,p. [ ] n Since Z n p is an n -dimensional vector space over Z p, gives the number of subgroups of dimension (rank) k, respectively, n,p gives the total number of subgroups, k p i.e. N(Z n p )= n,p. sing our key result, #Aut(Z p )=p 1 and Z n p = Z p Z n 1 p, that is L(Z p ) a chain with 2 elements, by counting the subgroups in the direct (lattices) product, respectively the diagonals, we obtain N(Z n p )=2N(Zn 1 p )+(p 1)n(1 seg(l(z n 1 p ))). Consequence. n(1 seg(l(z n p ))) = 1 p 1 [N(Zn+1 p ) 2N(Z n p )] = 1 p 1 [ n+1,p 2 n,p ]= pn 1 ence n(1 seg(l(z n p ))) = (p n 1 + p n p +1) n 1,p. p 1 n 1,p. Finally, we mention another straightforward recurrence one obtains by using the same key result : each finite p-group can be viewed as Z p l where is a (finite) direct sum of (finite) cocyclics, of order larger or equal to p l. If we know how to count the 1-segments, 2-segments,..., the l-segments in L( ) obviously the total number of subgroups is obtained adding (hereafter n(u seg) denotes the number of u-segments in ): l(p 1)n(1 seg) diagonals corresponding to isomorphisms Z p Z p ; +(l 1)p(p 1)n(2 seg) diagonals corresponding to isomorphisms Z p 2 Z p 2; +(l 2)p 2 (p 1)n(3 seg) diagonals corresponding to isomorphisms Z p 3 Z p 3;...
11 TE TOTAL NMBER OF SBROPS OF A FINITE ABELIAN ROP p l 2 (p 1)n((l 1) seg) diagonals corresponding to isomorphisms Z p l 1 Z p l 1; +p l 1 (p 1)n(l seg) diagonals corresponding to isomorphisms Z p l Z p l. ence N(Z p l )=l(p 1)n(1 seg)+(l 1)p(p 1)n(2 seg)+(l 2)p 2 (p 1)n(3 seg) p l 2 (p 1)n((l 1) seg) + p l 1 (p 1)n(l seg). References [1] Butler M. L., Subgroup Lattices and Symmetric Functions. Mem. AMS, vol. 112, nr. 539, [2] Delsarte S., Fonctions de Möbius sur les groupes abeliens finis. Annals of Math. 49, (1948) [3] Djubjuk P.E., On the number of subgroups of a finite abelian group. Izv. Akad. Nauk SSSR Ser. Mat. 12, (1948) [4] Frobenius., Stickelberger L., Über rupen von vertauschbaren Elementen. J. Reine und Angew. Math., 86 (1878), [5] Fuchs L., Infinite Abelian roups, vol. 1 and 2. Academic Press, New York, London 197, [6] oursat E., Sur les substitutions orthogonales et les divisions régulières de l espace. Ann. sci. école norm. sup (3) 6, (1889) [7] Macdonald I.., Symmetric Functions and all Polynomials. 2nd ed. Oxford Science Publications. Oxford: Clarendon Press, [8] Remak R., Über ntergruppen direkter Produkte von drei Faktoren. J. Reine Angew. Math. 166, (1931), [9] Schmidt R., Subgroup lattices of groups. de ruyter Expositions in Mathematics, 14. Walter de ruyter, [1] Steinitz E., Zur Theorie der Abel schen ruppen. Jahresbericht der DMV, 9, (191), 8-5. [11] Suzuki M., On the lattice of subgroups of finite groups. Trans. Amer. Math. Soc., 7, (1951) [12] Yeh Y., On prime power abelian groups. Bull. AMS, 54, (1948) uwait niversity, Faculty of Science, Dept. Sci., P.O. Box: 5969 Safat 136, State of uwait of Mathematics & Comp.
An arithmetic method of counting the subgroups of a finite abelian group
arxiv:180512158v1 [mathgr] 21 May 2018 An arithmetic method of counting the subgroups of a finite abelian group Marius Tărnăuceanu October 1, 2010 Abstract The main goal of this paper is to apply the arithmetic
More informationON THE SUBGROUPS OF FINITE ABELIAN GROUPS OF RANK THREE
Annales Univ. Sci. Budapest., Sect. Comp. 39 (2013) 111 124 ON THE SUBGROUPS OF FINITE ABELIAN GROUPS OF RANK THREE Mario Hampejs (Wien, Austria) László Tóth (Wien, Austria and Pécs, Hungary) Dedicated
More informationREMARKS ON THE PAPER SKEW PIERI RULES FOR HALL LITTLEWOOD FUNCTIONS BY KONVALINKA AND LAUVE
REMARKS ON THE PAPER SKEW PIERI RULES FOR HALL LITTLEWOOD FUNCTIONS BY KONVALINKA AND LAUVE S OLE WARNAAR Abstract In a recent paper Konvalinka and Lauve proved several skew Pieri rules for Hall Littlewood
More informationOn the number of diamonds in the subgroup lattice of a finite abelian group
DOI: 10.1515/auom-2016-0037 An. Şt. Univ. Ovidius Constanţa Vol. 24(2),2016, 205 215 On the number of diamonds in the subgroup lattice of a finite abelian group Dan Gregorian Fodor and Marius Tărnăuceanu
More informationABELIAN GROUPS WHOSE SUBGROUP LATTICE IS THE UNION OF TWO INTERVALS
J. Aust. Math. Soc. 78 (2005), 27 36 ABELIAN GROUPS WHOSE SUBGROUP LATTICE IS THE UNION OF TWO INTERVALS SIMION BREAZ and GRIGORE CĂLUGĂREANU (Received 15 February 2003; revised 21 July 2003) Communicated
More informationOn the number of cyclic subgroups of a finite Abelian group
Bull. Math. Soc. Sci. Math. Roumanie Tome 55103) No. 4, 2012, 423 428 On the number of cyclic subgroups of a finite Abelian group by László Tóth Abstract We prove by using simple number-theoretic arguments
More informationThe tensor algebra of power series spaces
The tensor algebra of power series spaces Dietmar Vogt Abstract The linear isomorphism type of the tensor algebra T (E) of Fréchet spaces and, in particular, of power series spaces is studied. While for
More informationSection II.2. Finitely Generated Abelian Groups
II.2. Finitely Generated Abelian Groups 1 Section II.2. Finitely Generated Abelian Groups Note. In this section we prove the Fundamental Theorem of Finitely Generated Abelian Groups. Recall that every
More informationADDITIVE GROUPS OF SELF-INJECTIVE RINGS
SOOCHOW JOURNAL OF MATHEMATICS Volume 33, No. 4, pp. 641-645, October 2007 ADDITIVE GROUPS OF SELF-INJECTIVE RINGS BY SHALOM FEIGELSTOCK Abstract. The additive groups of left self-injective rings, and
More informationOn the Frobenius Numbers of Symmetric Groups
Journal of Algebra 221, 551 561 1999 Article ID jabr.1999.7992, available online at http://www.idealibrary.com on On the Frobenius Numbers of Symmetric Groups Yugen Takegahara Muroran Institute of Technology,
More informationSubgroups of 3-factor direct products
Subgroups of 3-factor direct products Daniel Neuen and Pascal Schweitzer RWTH Aachen University {neuen,schweitzer}@informatik.rwth-aachen.de arxiv:1607.03444v1 [math.gr] 12 Jul 2016 July 13, 2016 Extending
More informationON THE SUM OF ELEMENT ORDERS OF FINITE ABELIAN GROUPS
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul...,..., f... DOI: 10.2478/aicu-2013-0013 ON THE SUM OF ELEMENT ORDERS OF FINITE ABELIAN GROUPS BY MARIUS TĂRNĂUCEANU and
More informationFinite groups determined by an inequality of the orders of their elements
Publ. Math. Debrecen 80/3-4 (2012), 457 463 DOI: 10.5486/PMD.2012.5168 Finite groups determined by an inequality of the orders of their elements By MARIUS TĂRNĂUCEANU (Iaşi) Abstract. In this note we introduce
More informationSylow 2-Subgroups of Solvable Q-Groups
E extracta mathematicae Vol. 22, Núm. 1, 83 91 (2007) Sylow 2-Subgroups of Solvable Q-roups M.R. Darafsheh, H. Sharifi Department of Mathematics, Statistics and Computer Science, Faculty of Science University
More informationAUTOMORPHISMS OF ORDER TWO*
AUTOMORPHISMS OF ORDER TWO* BY G. A. MILLER 1. Introduction. If an operator t transforms a group G into itself without being commutative with each operator of G, and if t2 is commutative with each operator
More informationMINIMAL NUMBER OF GENERATORS AND MINIMUM ORDER OF A NON-ABELIAN GROUP WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES
Communications in Algebra, 36: 1976 1987, 2008 Copyright Taylor & Francis roup, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870801941903 MINIMAL NUMBER OF ENERATORS AND MINIMUM ORDER OF
More informationNONABELIAN GROUPS WITH PERFECT ORDER SUBSETS
NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS CARRIE E. FINCH AND LENNY JONES Abstract. Let G be a finite group and let x G. Define the order subset of G determined by x to be the set of all elements in
More informationA proof of strong Goldbach conjecture and twin prime conjecture
A proof of strong Goldbach conjecture and twin prime conjecture Pingyuan Zhou E-mail:zhoupingyuan49@hotmail.com Abstract In this paper we give a proof of the strong Goldbach conjecture by studying limit
More informationISOMORPHISM OF MODULAR GROUP ALGEBRAS OF ABELIAN GROUPS WITH SEMI-COMPLETE p-primary COMPONENTS
Commun. Korean Math. Soc. 22 (2007), No. 2, pp. 157 161 ISOMORPHISM OF MODULAR GROUP ALGEBRAS OF ABELIAN GROUPS WITH SEMI-COMPLETE p-primary COMPONENTS Peter Danchev Reprinted from the Communications of
More informationarxiv: v2 [math.nt] 4 Jun 2016
ON THE p-adic VALUATION OF STIRLING NUMBERS OF THE FIRST KIND PAOLO LEONETTI AND CARLO SANNA arxiv:605.07424v2 [math.nt] 4 Jun 206 Abstract. For all integers n k, define H(n, k) := /(i i k ), where the
More informationON TYPES OF MATRICES AND CENTRALIZERS OF MATRICES AND PERMUTATIONS
ON TYPES OF MATRICES AND CENTRALIZERS OF MATRICES AND PERMUTATIONS JOHN R. BRITNELL AND MARK WILDON Abstract. It is known that that the centralizer of a matrix over a finite field depends, up to conjugacy,
More informationON A PROBLEM OF GEVORKYAN FOR THE FRANKLIN SYSTEM. Zygmunt Wronicz
Opuscula Math. 36, no. 5 (2016), 681 687 http://dx.doi.org/10.7494/opmath.2016.36.5.681 Opuscula Mathematica ON A PROBLEM OF GEVORKYAN FOR THE FRANKLIN SYSTEM Zygmunt Wronicz Communicated by P.A. Cojuhari
More informationTRIPLE FACTORIZATION OF NON-ABELIAN GROUPS BY TWO MAXIMAL SUBGROUPS
Journal of Algebra and Related Topics Vol. 2, No 2, (2014), pp 1-9 TRIPLE FACTORIZATION OF NON-ABELIAN GROUPS BY TWO MAXIMAL SUBGROUPS A. GHARIBKHAJEH AND H. DOOSTIE Abstract. The triple factorization
More informationNOTES Edited by William Adkins. On Goldbach s Conjecture for Integer Polynomials
NOTES Edited by William Adkins On Goldbach s Conjecture for Integer Polynomials Filip Saidak 1. INTRODUCTION. We give a short proof of the fact that every monic polynomial f (x) in Z[x] can be written
More informationTwo-boundary lattice paths and parking functions
Two-boundary lattice paths and parking functions Joseph PS Kung 1, Xinyu Sun 2, and Catherine Yan 3,4 1 Department of Mathematics, University of North Texas, Denton, TX 76203 2,3 Department of Mathematics
More informationTwo generator 4-Engel groups
Two generator 4-Engel groups Gunnar Traustason Centre for Mathematical Sciences Lund University Box 118, SE-221 00 Lund Sweden email: gt@maths.lth.se Using known results on 4-Engel groups one can see that
More informationPage Points Possible Points. Total 200
Instructions: 1. The point value of each exercise occurs adjacent to the problem. 2. No books or notes or calculators are allowed. Page Points Possible Points 2 20 3 20 4 18 5 18 6 24 7 18 8 24 9 20 10
More informationTILING ABELIAN GROUPS WITH A SINGLE TILE
TILING ABELIAN GROUPS WITH A SINGLE TILE S.J. EIGEN AND V.S. PRASAD Abstract. Suppose G is an infinite Abelian group that factorizes as the direct sum G = A B: i.e., the B-translates of the single tile
More informationGeneralization of a few results in integer partitions
Notes on Number Theory and Discrete Mathematics Vol. 9, 203, No. 2, 69 76 Generalization of a few results in integer partitions Manosij Ghosh Dastidar and Sourav Sen Gupta 2, Ramakrishna Mission Vidyamandira,
More informationKOREKTURY wim11203.tex
KOREKTURY wim11203.tex 23.12. 2005 REGULAR SUBMODULES OF TORSION MODULES OVER A DISCRETE VALUATION DOMAIN, Bandung,, Würzburg (Received July 10, 2003) Abstract. A submodule W of a p-primary module M of
More informationNovember In the course of another investigation we came across a sequence of polynomials
ON CERTAIN PLANE CURVES WITH MANY INTEGRAL POINTS F. Rodríguez Villegas and J. F. Voloch November 1997 0. In the course of another investigation we came across a sequence of polynomials P d Z[x, y], such
More informationA Note on Subgroup Coverings of Finite Groups
Analele Universităţii de Vest, Timişoara Seria Matematică Informatică XLIX, 2, (2011), 129 135 A Note on Subgroup Coverings of Finite Groups Marius Tărnăuceanu Abstract. In this note we determine the finite
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationTHE LIND-LEHMER CONSTANT FOR 3-GROUPS. Stian Clem 1 Cornell University, Ithaca, New York
#A40 INTEGERS 18 2018) THE LIND-LEHMER CONSTANT FOR 3-GROUPS Stian Clem 1 Cornell University, Ithaca, New York sac369@cornell.edu Christopher Pinner Department of Mathematics, Kansas State University,
More informationARTIN S CONJECTURE AND SYSTEMS OF DIAGONAL EQUATIONS
ARTIN S CONJECTURE AND SYSTEMS OF DIAGONAL EQUATIONS TREVOR D. WOOLEY Abstract. We show that Artin s conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of
More informationHamburger Beiträge zur Mathematik
Hamburger Beiträge zur Mathematik Nr. 270 / April 2007 Ernst Kleinert On the Restriction and Corestriction of Algebras over Number Fields On the Restriction and Corestriction of Algebras over Number Fields
More informationNILPOTENT NUMBERS JONATHAN PAKIANATHAN AND KRISHNAN SHANKAR
NILPOTENT NUMBERS JONATHAN PAKIANATHAN AND KRISHNAN SHANKAR Introduction. One of the first things we learn in abstract algebra is the notion of a cyclic group. For every positive integer n, we have Z n,
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationRANK AND PERIOD OF PRIMES IN THE FIBONACCI SEQUENCE. A TRICHOTOMY
RANK AND PERIOD OF PRIMES IN THE FIBONACCI SEQUENCE. A TRICHOTOMY Christian Ballot Université de Caen, Caen 14032, France e-mail: ballot@math.unicaen.edu Michele Elia Politecnico di Torino, Torino 10129,
More informationGraphs and Classes of Finite Groups
Note di Matematica ISSN 1123-2536, e-issn 1590-0932 Note Mat. 33 (2013) no. 1, 89 94. doi:10.1285/i15900932v33n1p89 Graphs and Classes of Finite Groups A. Ballester-Bolinches Departament d Àlgebra, Universitat
More informationC-Characteristically Simple Groups
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 35(1) (2012), 147 154 C-Characteristically Simple Groups M. Shabani Attar Department
More informationThe primitive root theorem
The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under
More informationarxiv: v1 [math.rt] 11 Sep 2009
FACTORING TILTING MODULES FOR ALGEBRAIC GROUPS arxiv:0909.2239v1 [math.rt] 11 Sep 2009 S.R. DOTY Abstract. Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of
More informationON VARIETIES IN WHICH SOLUBLE GROUPS ARE TORSION-BY-NILPOTENT
ON VARIETIES IN WHICH SOLUBLE GROUPS ARE TORSION-BY-NILPOTENT GÉRARD ENDIMIONI C.M.I., Université de Provence, UMR-CNRS 6632 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France E-mail: endimion@gyptis.univ-mrs.fr
More informationRELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS. Communicated by Ali Reza Ashrafi. 1. Introduction
Bulletin of the Iranian Mathematical Society Vol. 39 No. 4 (2013), pp 663-674. RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS A. ERFANIAN AND B. TOLUE Communicated by Ali Reza Ashrafi Abstract. Suppose
More informationThe power graph of a finite group, II
The power graph of a finite group, II Peter J. Cameron School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS, U.K. Abstract The directed power graph of a group G
More informationPermutation groups/1. 1 Automorphism groups, permutation groups, abstract
Permutation groups Whatever you have to do with a structure-endowed entity Σ try to determine its group of automorphisms... You can expect to gain a deep insight into the constitution of Σ in this way.
More informationAutomorphism Groups Definition. An automorphism of a group G is an isomorphism G G. The set of automorphisms of G is denoted Aut G.
Automorphism Groups 9-9-2012 Definition. An automorphism of a group G is an isomorphism G G. The set of automorphisms of G is denoted Aut G. Example. The identity map id : G G is an automorphism. Example.
More informationFinite groups determined by an inequality of the orders of their subgroups
Finite groups determined by an inequality of the orders of their subgroups Tom De Medts Marius Tărnăuceanu November 5, 2007 Abstract In this article we introduce and study two classes of finite groups
More informationarxiv: v1 [math.gr] 31 May 2016
FINITE GROUPS OF THE SAME TYPE AS SUZUKI GROUPS SEYED HASSAN ALAVI, ASHRAF DANESHKHAH, AND HOSEIN PARVIZI MOSAED arxiv:1606.00041v1 [math.gr] 31 May 2016 Abstract. For a finite group G and a positive integer
More informationGenerating Functions of Partitions
CHAPTER B Generating Functions of Partitions For a complex sequence {α n n 0,, 2, }, its generating function with a complex variable q is defined by A(q) : α n q n α n [q n ] A(q). When the sequence has
More informationCommunications in Algebra Publication details, including instructions for authors and subscription information:
This article was downloaded by: [Professor Alireza Abdollahi] On: 04 January 2013, At: 19:35 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered
More informationBOUNDS FOR ARITHMETIC HYPERBOLIC REFLECTION GROUPS IN DIMENSION 2
BOUNDS FOR ARITHMETIC HYPERBOLIC REFLECTION GROUPS IN DIMENSION 2 BENJAMIN LINOWITZ ABSTRACT. The work of Nikulin and Agol, Belolipetsky, Storm, and Whyte shows that only finitely many number fields may
More informationA connection between number theory and linear algebra
A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.
More informationRapporto di Ricerca CS G. Busetto, E. Jabara
UNIVERSITÀ CA FOSCARI DI VENEZIA Dipartimento di Informatica Technical Report Series in Computer Science Rapporto di Ricerca CS-2005-12 Ottobre 2005 G. Busetto, E. Jabara Some observations on factorized
More informationANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS Mathematics Subject Classification: 11B05, 11B13, 11P99
ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS N. HEGYVÁRI, F. HENNECART AND A. PLAGNE Abstract. We study the gaps in the sequence of sums of h pairwise distinct elements
More informationFuchs Problem When Torsion-Free Abelian Rank-One Groups are Slender
Irish Math. Soc. Bulletin 64 (2009), 79 83 79 Fuchs Problem When Torsion-Free Abelian Rank-One Groups are Slender PAVLOS TZERMIAS Abstract. We combine Baer s classification in [Duke Math. J. 3 (1937),
More informationThe Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete
The Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete Kyle Riggs Indiana University April 29, 2014 Kyle Riggs Decomposability Problem 1/ 22 Computable Groups background A group
More information#A69 INTEGERS 13 (2013) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES
#A69 INTEGERS 3 (203) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Greg Martin Department of
More informationTHE CLASSIFICATION OF INFINITE ABELIAN GROUPS WITH PARTIAL DECOMPOSITION BASES IN L ω
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number 2, 2017 THE CLASSIFICATION OF INFINITE ABELIAN GROUPS WITH PARTIAL DECOMPOSITION BASES IN L ω CAROL JACOBY AND PETER LOTH ABSTRACT. We consider the
More informationVladimir Kirichenko and Makar Plakhotnyk
São Paulo Journal of Mathematical Sciences 4, 1 (2010), 109 120 Gorenstein Quivers Vladimir Kirichenko and Makar Plakhotnyk Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University,
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] For
More information#A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC
#A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC Lenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania lkjone@ship.edu Received: 9/17/10, Revised:
More informationFIXED-POINT FREE ENDOMORPHISMS OF GROUPS RELATED TO FINITE FIELDS
FIXED-POINT FREE ENDOMORPHISMS OF GROUPS RELATED TO FINITE FIELDS LINDSAY N. CHILDS Abstract. Let G = F q β be the semidirect product of the additive group of the field of q = p n elements and the cyclic
More informationBY MolmS NEWMAN (1011) (11
THE STRUCTURE OF SOME SUBGROUPS OF THE MODULAR GROUP BY MolmS NEWMAN Introduction Let I be the 2 X 2 modular group. In a recent article [7] the notion of the type of a subgroup A of r was introduced. If
More informationOn the structure of some modules over generalized soluble groups
On the structure of some modules over generalized soluble groups L.A. Kurdachenko, I.Ya. Subbotin and V.A. Chepurdya Abstract. Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite
More informationgarcia de galdeano PRE-PUBLICACIONES del seminario matematico 2002 n. 16 garcía de galdeano Universidad de Zaragoza L.A. Kurdachenko J.
PRE-PUBLICACIONES del seminario matematico 2002 Groups with a few non-subnormal subgroups L.A. Kurdachenko J. Otal garcia de galdeano n. 16 seminario matemático garcía de galdeano Universidad de Zaragoza
More informationSPINNING AND BRANCHED CYCLIC COVERS OF KNOTS. 1. Introduction
SPINNING AND BRANCHED CYCLIC COVERS OF KNOTS C. KEARTON AND S.M.J. WILSON Abstract. A necessary and sufficient algebraic condition is given for a Z- torsion-free simple q-knot, q >, to be the r-fold branched
More informationA generalization of the Euler s totient function
arxiv:1312.1428v1 [math.gr] 5 Dec 2013 A generalization of the Euler s totient function Marius Tărnăuceanu December 5, 2013 Abstract The main goal of this paper is to provide a group theoretical generalization
More informationarxiv: v1 [math.ac] 10 Oct 2014
A BAER-KAPLANSKY THEOREM FOR MODULES OVER PRINCIPAL IDEAL DOMAINS arxiv:1410.2667v1 [math.ac] 10 Oct 2014 SIMION BREAZ Abstract. We will prove that if G H are modules over a principal ideal domain R such
More informationEuler characteristic of the truncated order complex of generalized noncrossing partitions
Euler characteristic of the truncated order complex of generalized noncrossing partitions D. Armstrong and C. Krattenthaler Department of Mathematics, University of Miami, Coral Gables, Florida 33146,
More informationASYMPTOTIC DIOPHANTINE APPROXIMATION: THE MULTIPLICATIVE CASE
ASYMPTOTIC DIOPHANTINE APPROXIMATION: THE MULTIPLICATIVE CASE MARTIN WIDMER ABSTRACT Let α and β be irrational real numbers and 0 < ε < 1/30 We prove a precise estimate for the number of positive integers
More informationHouston Journal of Mathematics. c 2007 University of Houston Volume 33, No. 1, 2007
Houston Journal of Mathematics c 2007 University of Houston Volume 33, No. 1, 2007 ROUPS WITH SPECIFIC NUMBER OF CENTRALIZERS A. ABDOLLAHI, S.M. JAFARIAN AMIRI AND A. MOHAMMADI HASSANABADI Communicated
More informationMaximal non-commuting subsets of groups
Maximal non-commuting subsets of groups Umut Işık March 29, 2005 Abstract Given a finite group G, we consider the problem of finding the maximal size nc(g) of subsets of G that have the property that no
More informationA Simple Classification of Finite Groups of Order p 2 q 2
Mathematics Interdisciplinary Research (017, xx yy A Simple Classification of Finite Groups of Order p Aziz Seyyed Hadi, Modjtaba Ghorbani and Farzaneh Nowroozi Larki Abstract Suppose G is a group of order
More informationON THE SUBGROUP LATTICE OF AN ABELIAN FINITE GROUP
ON THE SUBGROUP LATTICE OF AN ABELIAN FINITE GROUP Marius Tărnăuceanu Faculty of Mathematics Al.I. Cuza University of Iaşi, Romania e-mail: mtarnauceanu@yahoo.com The aim of this paper is to give some
More informationTripotents: a class of strongly clean elements in rings
DOI: 0.2478/auom-208-0003 An. Şt. Univ. Ovidius Constanţa Vol. 26(),208, 69 80 Tripotents: a class of strongly clean elements in rings Grigore Călugăreanu Abstract Periodic elements in a ring generate
More informationGROUPS WITH A MAXIMAL IRREDUNDANT 6-COVER #
Communications in Algebra, 33: 3225 3238, 2005 Copyright Taylor & Francis, Inc. ISSN: 0092-7872 print/1532-4125 online DOI: 10.1081/AB-200066157 ROUPS WITH A MAXIMAL IRREDUNDANT 6-COVER # A. Abdollahi,
More informationarxiv:math/ v1 [math.fa] 26 Oct 1993
arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological
More information1 p mr. r, where p 1 < p 2 < < p r are primes, reduces then to the problem of finding, for i = 1,...,r, all possible partitions (p e 1
Theorem 2.9 (The Fundamental Theorem for finite abelian groups). Let G be a finite abelian group. G can be written as an internal direct sum of non-trival cyclic groups of prime power order. Furthermore
More informationMINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS
MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RŮŽIČKA Abstract. A subset X of a group (or a ring, or a field) is called generating, if the smallest subgroup
More informationSmith theory. Andrew Putman. Abstract
Smith theory Andrew Putman Abstract We discuss theorems of P. Smith and Floyd connecting the cohomology of a simplicial complex equipped with an action of a finite p-group to the cohomology of its fixed
More informationLocally primitive normal Cayley graphs of metacyclic groups
Locally primitive normal Cayley graphs of metacyclic groups Jiangmin Pan Department of Mathematics, School of Mathematics and Statistics, Yunnan University, Kunming 650031, P. R. China jmpan@ynu.edu.cn
More informationDISTINGUISHING CARTESIAN PRODUCTS OF COUNTABLE GRAPHS
Discussiones Mathematicae Graph Theory 37 (2017) 155 164 doi:10.7151/dmgt.1902 DISTINGUISHING CARTESIAN PRODUCTS OF COUNTABLE GRAPHS Ehsan Estaji Hakim Sabzevari University, Sabzevar, Iran e-mail: ehsan.estaji@hsu.ac.ir
More informationCHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSAL-EXISTENTIAL FORMULA
CHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSAL-EXISTENTIAL FORMULA BJORN POONEN Abstract. We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential
More informationNESTED SYMMETRIC REPRESENTATION OF ELEMENTS OF THE SUZUKI CHAIN GROUPS
IJMMS 2003:62, 3931 3948 PII. S0161171203212023 http://ijmms.hindawi.com Hindawi Publishing Corp. NESTED SYMMETRIC REPRESENTATION OF ELEMENTS OF THE SUZUKI CHAIN GROUPS MOHAMED SAYED Received 2 December
More informationA GENERALIZATION OF DAVENPORT S CONSTANT AND ITS ARITHMETICAL APPLICATIONS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXIII 1992 FASC. 2 A GENERALIZATION OF DAVENPORT S CONSTANT AND ITS ARITHMETICAL APPLICATIONS BY FRANZ H A L T E R - K O C H (GRAZ) 1. For an additively
More informationMaximal perpendicularity in certain Abelian groups
Acta Univ. Sapientiae, Mathematica, 9, 1 (2017) 235 247 DOI: 10.1515/ausm-2017-0016 Maximal perpendicularity in certain Abelian groups Mika Mattila Department of Mathematics, Tampere University of Technology,
More informationCAYLEY NUMBERS WITH ARBITRARILY MANY DISTINCT PRIME FACTORS arxiv: v1 [math.co] 17 Sep 2015
CAYLEY NUMBERS WITH ARBITRARILY MANY DISTINCT PRIME FACTORS arxiv:1509.05221v1 [math.co] 17 Sep 2015 TED DOBSON AND PABLO SPIGA Abstract. A positive integer n is a Cayley number if every vertex-transitive
More informationOdd-order Cayley graphs with commutator subgroup of order pq are hamiltonian
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 1 28 Odd-order Cayley graphs with commutator subgroup of order
More informationUNIMODALITY OF PARTITIONS WITH DISTINCT PARTS INSIDE FERRERS SHAPES
UNIMODALITY OF PARTITIONS WITH DISTINCT PARTS INSIDE FERRERS SHAPES RICHARD P. STANLEY AND FABRIZIO ZANELLO Abstract. We investigate the rank-generating function F λ of the poset of partitions contained
More informationOn the image of noncommutative local reciprocity map
On the image of noncommutative local reciprocity map Ivan Fesenko 0. Introduction First steps in the direction of an arithmetic noncommutative local class field theory were described in [] as an attempt
More informationON SOME FACTORIZATION FORMULAS OF K-k-SCHUR FUNCTIONS
ON SOME FACTORIZATION FORMULAS OF K-k-SCHUR FUNCTIONS MOTOKI TAKIGIKU Abstract. We give some new formulas about factorizations of K-k-Schur functions, analogous to the k-rectangle factorization formula
More informationELEMENTARY GROUPS BY HOMER BECHTELL
ELEMENTARY GROUPS BY HOMER BECHTELL 1. Introduction. The purpose of this paper is to investigate two classes of finite groups, elementary groups and E-groups. The elementary groups have the property that
More informationNon-separable AF-algebras
Non-separable AF-algebras Takeshi Katsura Department of Mathematics, Hokkaido University, Kita 1, Nishi 8, Kita-Ku, Sapporo, 6-81, JAPAN katsura@math.sci.hokudai.ac.jp Summary. We give two pathological
More informationMULTI-ORDERED POSETS. Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A06 MULTI-ORDERED POSETS Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States lbishop@oxy.edu
More informationR E N D I C O N T I PRIME GRAPH COMPONENTS OF FINITE ALMOST SIMPLE GROUPS
R E N D I C O N T I del Seminario Matematico dell Università di Padova Vol. 102 Anno 1999 PRIME GRAPH COMPONENTS OF FINITE ALMOST SIMPLE GROUPS Maria Silvia Lucido Dipartimento di Matematica Pura e Applicata
More informationProperties of Generating Sets of Finite Groups
Cornell SPUR 2018 1 Group Theory Properties of Generating Sets of Finite Groups by R. Keith Dennis We now provide a few more details about the prerequisites for the REU in group theory, where to find additional
More informationZAREMBA'S CONJECTURE AND SUMS OF THE DIVISOR FUNCTION
mathematics of computation volume 61, number 203 july 1993, pages 171-176 ZAREMBA'S CONJECTURE AND SUMS OF THE DIVISOR FUNCTION T. W. CUSICK Dedicated to the memory ofd. H. Lehmer Abstract. Zaremba conjectured
More informationDegree Graphs of Simple Orthogonal and Symplectic Groups
Degree Graphs of Simple Orthogonal and Symplectic Groups Donald L. White Department of Mathematical Sciences Kent State University Kent, Ohio 44242 E-mail: white@math.kent.edu Version: July 12, 2005 In
More information